We use the Two-Degree Field Galaxy Redshift Survey (2dFGRS) to test the hierarchical scaling hypothesis: namely, that the p-point galaxy correlation functions can be written in terms of the two-point correlation function or variance. This scaling is expected if an initially Gaussian distribution of density fluctuations evolves under the action of gravitational instability. We measure the volume-averaged p-point correlation functions using a counts-in-cells technique applied to a volume-limited sample of 44 931 L* galaxies. We demonstrate that L* galaxies display hierarchical clustering up to order p = 6 in redshift space. The variance measured for L* galaxies is in excellent agreement with the predictions from a Λ-cold dark matter N-body simulation. This applies to all cell radii considered, 0.3 < (R/h-1 Mpc) < 30. However, the higher order correlation functions of L* galaxies have a significantly smaller amplitude than is predicted for the dark matter for R < 10 h-1 Mpc. This disagreement implies that a non-linear bias exists between the dark matter and L* galaxies on these scales. We also show that the presence of two rare, massive superclusters in the 2dFGRS has an impact on the higher-order clustering moments measured on large scales.